Hausdorff dimension and projections related to intersections
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Pertti Mattila
University of Helsinki. Department of Mathematics and Statistics
For Sg(x, y) = x − g(y), x, y ∈ Rn, g ∈ O(n), we investigate the Lebesgue
measure and Hausdorff dimension of Sg(A) given the dimension of A, both for general
Borel subsets of R2n and for product sets.
Paraules clau
Hausdorff dimension, projectio, intersection, Fourier transform
Article Details
Com citar
Mattila, Pertti. «Hausdorff dimension and projections related to intersections». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 1, p. 305-23, http://raco.cat/index.php/PublicacionsMatematiques/article/view/396522.
Referències
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measures on hyperboloids, Trans. Amer. Math. Soc. 374(2) (2021), 1041–1075.
DOI: 10.1090/tran/8283.
C.-H. Cho, S. Ham, and S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal. 150 (2017), 61–75. DOI: 10.1016/j.na.2016.11.006.
X. Du, Upper bounds for Fourier decay rates of fractal measures, J. Lond. Math.
Soc. (2) 102(3) (2020), 1318–1336. DOI: 10.1112/jlms.12364.
X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, and R. Zhang, Weighted
restriction estimates and application to Falconer distance set problem, Amer. J.
Math. 143(1) (2021), 175–211. DOI: 10.1353/ajm.2021.0005.
X. Du, A. Iosevich, Y. Ou, H. Wang, and R. Zhang, An improved result
for Falconer’s distance set problem in even dimensions, Math. Ann. 380(3–4)
(2021), 1215–1231. DOI: 10.1007/s00208-021-02170-1.
X. Du and R. Zhang, Sharp L2 estimates of the Schr¨odinger maximal function
in higher dimensions, Ann. of Math. (2) 189(3) (2019), 837–861. DOI: 10.4007/
annals.2019.189.3.4.
M. B. Erdogan, A note on the Fourier transform of fractal measures, Math.
Res. Lett. 11(2–3) (2004), 299–313. DOI: 10.4310/MRL.2004.v11.n3.a3.
S. Eswarathasan, A. Iosevich, and K. Taylor, Intersections of sets and
Fourier analysis, J. Anal. Math. 128 (2016), 159–178. DOI: 10.1007/s11854-016
-0004-1.
K. J. Falconer, Hausdorff dimension and the exceptional set of projections,
Mathematika 29(1) (1982), 109–115. DOI: 10.1112/S0025579300012201.
K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika
32(2) (1985), 206–212 (1986). DOI: 10.1112/S0025579300010998.
[11] K. Fassler and T. Orponen ¨ , On restricted families of projections in R3, Proc.
Lond. Math. Soc. (3) 109(2) (2014), 353–381. DOI: 10.1112/plms/pdu004.
L. Guth, A. Iosevich, Y. Ou, and H. Wang, On Falconer’s distance set
problem in the plane, Invent. Math. 219(3) (2020), 779–830. DOI: 10.1007/
s00222-019-00917-x.
T. L. J. Harris, Improved decay of conical averages of the Fourier transform,
Proc. Amer. Math. Soc. 147(11) (2019), 4781–4796. DOI: 10.1090/proc/14747.
T. L. J. Harris, Restricted families of projections onto planes: the general case
of nonvanishing geodesic curvature, Preprint (2021). arXiv:2107.14701.
A. Iosevich and B. Liu, Falconer distance problem, additive energy and
Cartesian products, Ann. Acad. Sci. Fenn. Math. 41(2) (2016), 579–585.
DOI: 10.5186/aasfm.2016.4135.
E. Järvenpää, M. Järvenpää, and T. Keleti ¨ , Hausdorff dimension and nondegenerate families of projections, J. Geom. Anal. 24(4) (2014), 2020–2034.
DOI: 10.1007/s12220-013-9407-8.
E. Järvenpää, M. Järvenpää, F. Ledrappier, and M. Leikas, One-dimensional families of projections, Nonlinearity 21(3) (2008), 453–463. DOI: 10.1088/
0951-7715/21/3/005.
M. Järvenpää, On the upper Minkowski dimension, the packing dimension, and
orthogonal projections, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 99
(1994), 34 pp.
M. Järvenpää, Concerning the packing dimension of intersection measures,
Math. Proc. Cambridge Philos. Soc. 121(2) (1997), 287–296. DOI: 10.1017/
S0305004196001417.
M. Järvenpää, Packing dimension, intersection measures, and isometries,
Math. Proc. Cambridge Philos. Soc. 122(3) (1997), 483–490. DOI: 10.1017/
S0305004197001941.
A. Käenmäki, T. Orponen, and L. Venieri, A Marstrand-type restricted projection theorem in R3, Preprint (2017). arXiv:1708.04859.
J.-P. Kahane, Sur la dimension des intersections, in: “Aspects of Mathematics
and its Applications”, North-Holland Math. Library 34, North-Holland, Amsterdam, 1986, pp. 419–430. DOI: 10.1016/S0924-6509(09)70272-7.
R. Kaufman, On Hausdorff dimension of projections, Mathematika 15(2)
(1968), 153–155. DOI: 10.1112/S0025579300002503.
R. Kaufman and P. Mattila, Hausdorff dimension and exceptional sets of linear transformations, Ann. Acad. Sci. Fenn. Ser. A I Math. 1(2) (1975), 387–392.
DOI: 10.5186/AASFM.1975.0105.
J. M. Marstrand, Some fundamental geometrical properties of plane sets
of fractional dimensions, Proc. London Math. Soc. (3) 4(1) (1954), 257–302.
DOI: 10.1112/plms/s3-4.1.257.
P. Mattila, Hausdorff dimension and capacities of intersections of sets in nspace, Acta Math. 152(1–2) (1984), 77–105. DOI: 10.1007/BF02392192.
P. Mattila, On the Hausdorff dimension and capacities of intersections, Mathematika 32(2) (1985), 213–217 (1986). DOI: 10.1112/S0025579300011001.
P. Mattila, Spherical averages of Fourier transforms of measures with finite
energy; dimension of intersections and distance sets, Mathematika 34(2) (1987),
207–228. DOI: 10.1112/S0025579300013462.
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Fractals
and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge
University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813.
P. Mattila, “Fourier Analysis and Hausdorff Dimension”, Cambridge Studies
in Advanced Mathematics 150, Cambridge University Press, Cambridge, 2015.
DOI: 10.1017/CBO9781316227619.
P. Mattila, Exceptional set estimates for the Hausdorff dimension of intersections, Ann. Acad. Sci. Fenn. Math. 42(2) (2017), 611–620. DOI: 10.5186/aasfm.
2017.4236.
P. Mattila, Hausdorff Dimension, Projections, Intersections, and Besicovitch
Sets, in: “New Trends in Applied Harmonic Analysis”, Volume 2, Applied and
Numerical Harmonic Analysis, Birkhäuser, Cham, 2019, pp. 129–157. DOI: 10.
1007/978-3-030-32353-0_6.
P. Mattila, Hausdorff dimension of intersections with planes and general sets,
Preprint (2020). arXiv:2005.11790.
D. M. Oberlin, Exceptional sets of projections, unions of k-planes and associated transforms, Israel J. Math. 202(1) (2014), 331–342. DOI: 10.1007/
s11856-014-1040-4.
D. Oberlin and R. Oberlin, Application of a Fourier restriction theorem to
certain families of projections in R3, J. Geom. Anal. 25(3) (2015), 1476–1491.
DOI: 10.1007/s12220-014-9480-7.
T. Orponen, Hausdorff dimension estimates for restricted families of projections
in R3, Adv. Math. 275 (2015), 147–183. DOI: 10.1016/j.aim.2015.02.011.
T. Orponen and L. Venieri, Improved bounds for restricted families of projections to planes in R3, Int. Math. Res. Not. IMRN 2020(19) (2020), 5797–5813.
DOI: 10.1093/imrn/rny193.
T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 1999(10) (1999), 547–567. DOI: 10.1155/S1073792899
000288.
measures on hyperboloids, Trans. Amer. Math. Soc. 374(2) (2021), 1041–1075.
DOI: 10.1090/tran/8283.
C.-H. Cho, S. Ham, and S. Lee, Fractal Strichartz estimate for the wave equation, Nonlinear Anal. 150 (2017), 61–75. DOI: 10.1016/j.na.2016.11.006.
X. Du, Upper bounds for Fourier decay rates of fractal measures, J. Lond. Math.
Soc. (2) 102(3) (2020), 1318–1336. DOI: 10.1112/jlms.12364.
X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, and R. Zhang, Weighted
restriction estimates and application to Falconer distance set problem, Amer. J.
Math. 143(1) (2021), 175–211. DOI: 10.1353/ajm.2021.0005.
X. Du, A. Iosevich, Y. Ou, H. Wang, and R. Zhang, An improved result
for Falconer’s distance set problem in even dimensions, Math. Ann. 380(3–4)
(2021), 1215–1231. DOI: 10.1007/s00208-021-02170-1.
X. Du and R. Zhang, Sharp L2 estimates of the Schr¨odinger maximal function
in higher dimensions, Ann. of Math. (2) 189(3) (2019), 837–861. DOI: 10.4007/
annals.2019.189.3.4.
M. B. Erdogan, A note on the Fourier transform of fractal measures, Math.
Res. Lett. 11(2–3) (2004), 299–313. DOI: 10.4310/MRL.2004.v11.n3.a3.
S. Eswarathasan, A. Iosevich, and K. Taylor, Intersections of sets and
Fourier analysis, J. Anal. Math. 128 (2016), 159–178. DOI: 10.1007/s11854-016
-0004-1.
K. J. Falconer, Hausdorff dimension and the exceptional set of projections,
Mathematika 29(1) (1982), 109–115. DOI: 10.1112/S0025579300012201.
K. J. Falconer, On the Hausdorff dimensions of distance sets, Mathematika
32(2) (1985), 206–212 (1986). DOI: 10.1112/S0025579300010998.
[11] K. Fassler and T. Orponen ¨ , On restricted families of projections in R3, Proc.
Lond. Math. Soc. (3) 109(2) (2014), 353–381. DOI: 10.1112/plms/pdu004.
L. Guth, A. Iosevich, Y. Ou, and H. Wang, On Falconer’s distance set
problem in the plane, Invent. Math. 219(3) (2020), 779–830. DOI: 10.1007/
s00222-019-00917-x.
T. L. J. Harris, Improved decay of conical averages of the Fourier transform,
Proc. Amer. Math. Soc. 147(11) (2019), 4781–4796. DOI: 10.1090/proc/14747.
T. L. J. Harris, Restricted families of projections onto planes: the general case
of nonvanishing geodesic curvature, Preprint (2021). arXiv:2107.14701.
A. Iosevich and B. Liu, Falconer distance problem, additive energy and
Cartesian products, Ann. Acad. Sci. Fenn. Math. 41(2) (2016), 579–585.
DOI: 10.5186/aasfm.2016.4135.
E. Järvenpää, M. Järvenpää, and T. Keleti ¨ , Hausdorff dimension and nondegenerate families of projections, J. Geom. Anal. 24(4) (2014), 2020–2034.
DOI: 10.1007/s12220-013-9407-8.
E. Järvenpää, M. Järvenpää, F. Ledrappier, and M. Leikas, One-dimensional families of projections, Nonlinearity 21(3) (2008), 453–463. DOI: 10.1088/
0951-7715/21/3/005.
M. Järvenpää, On the upper Minkowski dimension, the packing dimension, and
orthogonal projections, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 99
(1994), 34 pp.
M. Järvenpää, Concerning the packing dimension of intersection measures,
Math. Proc. Cambridge Philos. Soc. 121(2) (1997), 287–296. DOI: 10.1017/
S0305004196001417.
M. Järvenpää, Packing dimension, intersection measures, and isometries,
Math. Proc. Cambridge Philos. Soc. 122(3) (1997), 483–490. DOI: 10.1017/
S0305004197001941.
A. Käenmäki, T. Orponen, and L. Venieri, A Marstrand-type restricted projection theorem in R3, Preprint (2017). arXiv:1708.04859.
J.-P. Kahane, Sur la dimension des intersections, in: “Aspects of Mathematics
and its Applications”, North-Holland Math. Library 34, North-Holland, Amsterdam, 1986, pp. 419–430. DOI: 10.1016/S0924-6509(09)70272-7.
R. Kaufman, On Hausdorff dimension of projections, Mathematika 15(2)
(1968), 153–155. DOI: 10.1112/S0025579300002503.
R. Kaufman and P. Mattila, Hausdorff dimension and exceptional sets of linear transformations, Ann. Acad. Sci. Fenn. Ser. A I Math. 1(2) (1975), 387–392.
DOI: 10.5186/AASFM.1975.0105.
J. M. Marstrand, Some fundamental geometrical properties of plane sets
of fractional dimensions, Proc. London Math. Soc. (3) 4(1) (1954), 257–302.
DOI: 10.1112/plms/s3-4.1.257.
P. Mattila, Hausdorff dimension and capacities of intersections of sets in nspace, Acta Math. 152(1–2) (1984), 77–105. DOI: 10.1007/BF02392192.
P. Mattila, On the Hausdorff dimension and capacities of intersections, Mathematika 32(2) (1985), 213–217 (1986). DOI: 10.1112/S0025579300011001.
P. Mattila, Spherical averages of Fourier transforms of measures with finite
energy; dimension of intersections and distance sets, Mathematika 34(2) (1987),
207–228. DOI: 10.1112/S0025579300013462.
P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Fractals
and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge
University Press, Cambridge, 1995. DOI: 10.1017/CBO9780511623813.
P. Mattila, “Fourier Analysis and Hausdorff Dimension”, Cambridge Studies
in Advanced Mathematics 150, Cambridge University Press, Cambridge, 2015.
DOI: 10.1017/CBO9781316227619.
P. Mattila, Exceptional set estimates for the Hausdorff dimension of intersections, Ann. Acad. Sci. Fenn. Math. 42(2) (2017), 611–620. DOI: 10.5186/aasfm.
2017.4236.
P. Mattila, Hausdorff Dimension, Projections, Intersections, and Besicovitch
Sets, in: “New Trends in Applied Harmonic Analysis”, Volume 2, Applied and
Numerical Harmonic Analysis, Birkhäuser, Cham, 2019, pp. 129–157. DOI: 10.
1007/978-3-030-32353-0_6.
P. Mattila, Hausdorff dimension of intersections with planes and general sets,
Preprint (2020). arXiv:2005.11790.
D. M. Oberlin, Exceptional sets of projections, unions of k-planes and associated transforms, Israel J. Math. 202(1) (2014), 331–342. DOI: 10.1007/
s11856-014-1040-4.
D. Oberlin and R. Oberlin, Application of a Fourier restriction theorem to
certain families of projections in R3, J. Geom. Anal. 25(3) (2015), 1476–1491.
DOI: 10.1007/s12220-014-9480-7.
T. Orponen, Hausdorff dimension estimates for restricted families of projections
in R3, Adv. Math. 275 (2015), 147–183. DOI: 10.1016/j.aim.2015.02.011.
T. Orponen and L. Venieri, Improved bounds for restricted families of projections to planes in R3, Int. Math. Res. Not. IMRN 2020(19) (2020), 5797–5813.
DOI: 10.1093/imrn/rny193.
T. Wolff, Decay of circular means of Fourier transforms of measures, Internat. Math. Res. Notices 1999(10) (1999), 547–567. DOI: 10.1155/S1073792899
000288.